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arXiv · 2106.01996

A Jacobian criterion for nonsingularity in mixed characteristic

Melvin Hochster·Jack Jeffries
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Abstract

We give a version of the usual Jacobian characterization of the defining ideal of the singular locus in the equal characteristic case: the new theorem is valid for essentially affine algebras over a complete local algebra over a mixed characteristic discrete valuation ring. The result makes use of the minors of a matrix that includes a row coming from the values of a $p$-derivation. To study the analogue of modules of differentials associated with the mixed Jacobian matrices that arise in our context, we introduce and investigate the notion of a perivation, which may be thought of, roughly, as a linearization of the notion of $p$-derivation. We also develop a mixed characteristic analogue of the positive characteristic $\Gamma$-construction, and apply this to give additional nonsingularity criteria.

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Melvin Hochster, Jack Jeffries. 2021-06-03. A Jacobian criterion for nonsingularity in mixed characteristic. https://arxiv.org/abs/2106.01996

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